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Mirrors > Home > MPE Home > Th. List > prodrblem2 | Structured version Visualization version GIF version |
Description: Lemma for prodrb 15475. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodmo.1 | ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
prodmo.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
prodrb.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
prodrb.5 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
prodrb.6 | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
prodrb.7 | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) |
Ref | Expression |
---|---|
prodrblem2 | ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodrb.5 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) |
3 | seqex 13559 | . . 3 ⊢ seq𝑀( · , 𝐹) ∈ V | |
4 | climres 15119 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑀( · , 𝐹) ∈ V) → ((seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) ⇝ 𝐶 ↔ seq𝑀( · , 𝐹) ⇝ 𝐶)) | |
5 | 2, 3, 4 | sylancl 589 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) ⇝ 𝐶 ↔ seq𝑀( · , 𝐹) ⇝ 𝐶)) |
6 | prodrb.7 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) | |
7 | prodmo.1 | . . . . 5 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) | |
8 | prodmo.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
9 | 8 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
10 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
11 | 7, 9, 10 | prodrblem 15472 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( · , 𝐹)) |
12 | 6, 11 | mpidan 689 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( · , 𝐹)) |
13 | 12 | breq1d 5053 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((seq𝑀( · , 𝐹) ↾ (ℤ≥‘𝑁)) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
14 | 5, 13 | bitr3d 284 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ⊆ wss 3857 ifcif 4429 class class class wbr 5043 ↦ cmpt 5124 ↾ cres 5542 ‘cfv 6369 ℂcc 10710 1c1 10713 · cmul 10717 ℤcz 12159 ℤ≥cuz 12421 seqcseq 13557 ⇝ cli 15028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-seq 13558 df-clim 15032 |
This theorem is referenced by: prodrb 15475 |
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